“The second he said it, you could feel the air in the room change. Barack Obama’s warning that Brexit would mean “the UK is going to be at the back of the queue” of America’s trade priorities was as brutal as it was effective. And the sharp intake of breath from the gathered press pack, greeted by the knowingly satisfied smiles of Downing Street aides, was proof that the President had delivered and delivered in spades.”

So wrote Paul Waugh of the Huffington Post in his newsletter report of the President’s Downing Street press conference. Are those aides still smiling with satisfaction this morning?

We have now had four polls on the EU referendum since Obama’s visit, as follows:

In short, there is no suggestion from any of them that the President’s intervention has made any substantial difference to referendum voting intentions.

Now, it may be that all these polls are “wrong” – in other words, that Obama’s trip, and his view of any post-Brexit trade talks between America and Britain, are having an effect on voters’ opinions that they haven’t picked up.

But the most likely explanation is also the most simple. The overwhelming majority of British voters are not members of the Westminster Village, and consequently got less agitated than it about the President’s visit.

Indeed, very many will not have noticed it at all. Others may have glanced at a TV screen or glimpsed a headline. But in terms of grabbing their attention, the President was, to coin a phrase, “at the back of the queue”.

If so, there are two lessons in these findings. The first is about Britain and America (and Obama); the second is about this referendum and the polls.

Many people active on the Left dislike the United States, but like Obama. For many on the Right, it is the other way round. But both share a interest in American politics, amounting in some cases to a West Wing-flavoured passion.

Most voters are certainly less engaged with and arguably more cool about the United States, or at least the views of its politicians insofar as they touch on Britain.

Those breath-drawing lobby journalists and knowingly-smiling Number Ten aides are part of a group that, one way or the other, tends to get overwrought about America and its politics and overwhelmed by the glamour of its presidency.

On the referendum and the polls, we are still too far out from polling day for them to be more than “snapshots, not predictions”. Local elections take place a week today, and the state opening of Parliament on May 18.

When it is over, the Government’s decks will be cleared for the referendum run-in of about a month. Only then will most voters engage their minds more actively with the choice before them.